Problem 70 - College Algebra with Corequisite Support

Question

Calculate the first eight terms of the sequences \(a_{n}=\frac{(n+2) !}{(n-1) !}\) and \(b_{n}=n^{3}+3 n^{2}+2 n,\) and then make a conjecture about the relationship between these two sequences.

Step-by-Step Solution

Answer

The sequences are identical for the first eight terms, conjecture: \(a_n = b_n\).

Step 1: Understanding the sequences

We are given two different sequences: \(a_n = \frac{(n+2)!}{(n-1)!}\) and \(b_n = n^3 + 3n^2 + 2n\). Our goal is to find the first eight terms for each sequence.

Step 2: Calculating the first eight terms of \(a_n\)

To find the terms of sequence \(a_n\), we calculate: - \(a_1 = \frac{(1+2)!}{(1-1)!} = \frac{3!}{0!} = \frac{6}{1} = 6\)- \(a_2 = \frac{(2+2)!}{(2-1)!} = \frac{4!}{1!} = \frac{24}{1} = 24\)- \(a_3 = \frac{(3+2)!}{(3-1)!} = \frac{5!}{2!} = \frac{120}{2} = 60\)- \(a_4 = \frac{(4+2)!}{(4-1)!} = \frac{6!}{3!} = \frac{720}{6} = 120\)- \(a_5 = \frac{(5+2)!}{(5-1)!} = \frac{7!}{4!} = \frac{5040}{24} = 210\)- \(a_6 = \frac{(6+2)!}{(6-1)!} = \frac{8!}{5!} = \frac{40320}{120} = 336\)- \(a_7 = \frac{(7+2)!}{(7-1)!} = \frac{9!}{6!} = \frac{362880}{720} = 504\)- \(a_8 = \frac{(8+2)!}{(8-1)!} = \frac{10!}{7!} = \frac{3628800}{5040} = 720\)

Step 3: Calculating the first eight terms of \(b_n\)

For sequence \(b_n\), we compute each term as follows:- \(b_1 = 1^3 + 3\cdot1^2 + 2\cdot1 = 1 + 3 + 2 = 6\)- \(b_2 = 2^3 + 3\cdot2^2 + 2\cdot2 = 8 + 12 + 4 = 24\)- \(b_3 = 3^3 + 3\cdot3^2 + 2\cdot3 = 27 + 27 + 6 = 60\)- \(b_4 = 4^3 + 3\cdot4^2 + 2\cdot4 = 64 + 48 + 8 = 120\)- \(b_5 = 5^3 + 3\cdot5^2 + 2\cdot5 = 125 + 75 + 10 = 210\)- \(b_6 = 6^3 + 3\cdot6^2 + 2\cdot6 = 216 + 108 + 12 = 336\)- \(b_7 = 7^3 + 3\cdot7^2 + 2\cdot7 = 343 + 147 + 14 = 504\)- \(b_8 = 8^3 + 3\cdot8^2 + 2\cdot8 = 512 + 192 + 16 = 720\)

Step 4: Making a conjecture

Comparing the eight terms of both sequences, we observe that they are identical: 6, 24, 60, 120, 210, 336, 504, 720. Therefore, a reasonable conjecture is that \(a_n = b_n\) for all \(n\).

Key Concepts

Factorials

The factorial function is a fundamental concept in mathematics, especially in sequences and combinatorics. It is represented by an exclamation mark (!). The factorial of a number, denoted as \( n! \), is the product of all positive integers less than or equal to \( n \). For example, \( 4! = 4 \times 3 \times 2 \times 1 = 24 \). Factorials rapidly increase as \( n \) becomes large, which makes them suitable for calculations involving permutations and combinations.
To calculate factorial-based sequences, like in the exercise given, understanding the simplification of factorial expressions is essential. When calculating \( a_n = \frac{(n+2)!}{(n-1)!} \), notice how many terms cancel out:

• \( (n+2)! = (n+2) \times (n+1) \times n \times (n-1)! \)
• The factorial of \((n-1)!\) cancels with that in the denominator, leaving \((n+2) \times (n+1) \times n \) for the calculation of each term \( a_n \).

This simplification allows quicker computation, crucial for efficiently finding sequence terms.

Polynomial Functions

Polynomial functions are algebraic expressions consisting of variables raised to non-negative integer powers, coefficients, and constant terms. They are written in the form \( f(x) = a_nx^n + a_{n-1}x^{n-1} + \ldots + a_1x + a_0 \). In this exercise, the polynomial function is given by \( b_n = n^3 + 3n^2 + 2n \).
The sequence derived from a polynomial function often depicts a more predictable pattern than other complex functions. Here, the specific cubic polynomial is expressed by combining cubes of \( n \), squares of \( n \), and linear terms.
Key steps to understand polynomial sequences:

• Identify the degree: The highest power, 3 in our polynomial, determines the cubic nature of \( b_n \).
• Calculate each component separately: Breaking down the polynomial into its parts (e.g., \( n^3 \), \( 3n^2 \), and \( 2n \)) can aid in computation and understanding.

Ultimately, polynomial sequences are straightforward to compute and predict, exemplified by substituting integers into \( b_n \) to find its terms.

Conjecture Formulation

Conjecture formulation involves making educated guesses or hypotheses based on observed patterns or sequences without a formal proof. It plays a crucial role in mathematical exploration and discovery.
In the example given, after computing terms of two different sequences, you might notice they share the same values: 6, 24, 60, 120, 210, 336, 504, 720 for both \( a_n \) and \( b_n \). This pattern suggests a relation: despite differing forms, the sequences are equivalent term by term.

• Formulating this conjecture involves recognizing identical terms over the initial eight calculated terms.
• The conjecture \( a_n = b_n \) stems from these observations, paving the way for further exploration or formal proof to verify this pattern holds for larger values of \( n \).

Conjectures allow mathematicians to uncover deeper truths and prove or disprove proposed theories with formal mathematical proofs later.